Integrated sources of photon quantum states based on nonlinear optics

The ability to generate complex optical photon states involving entanglement between multiple optical modes is not only critical to advancing our understanding of quantum mechanics but will play a key role in generating many applications in quantum technologies. These include quantum communications, computation, imaging, microscopy and many other novel technologies that are constantly being proposed. However, approaches to generating parallel multiple, customisable bi- and multi-entangled quantum bits (qubits) on a chip are still in the early stages of development. Here, we review recent advances in the realisation of integrated sources of photonic quantum states, focusing on approaches based on nonlinear optics that are compatible with contemporary optical fibre telecommunications and quantum memory platforms as well as with chip-scale semiconductor technology. These new and exciting platforms hold the promise of compact, low-cost, scalable and practical implementations of sources for the generation and manipulation of complex quantum optical states on a chip, which will play a major role in bringing quantum technologies out of the laboratory and into the real world.


A: Definition of quantum superposition and quantum entanglement
Quantum superposition: If we consider, for example, a quantum state that can be described by a twodimensional space (such as a photon polarisation or an electron spin), we find that the quantum state can be described not only as being in state A (e.g., horizontal polarisation or spin up) or B (e.g., vertical polarisation or spin down) but also in a superposition of the two: where α and β are two complex parameters related by the normalisation condition | | 2 + | | 2 = 1. Note that this is conceptually different from the scenario where a system is either in state A with probability | | 2 or in state B with probability | | 2 (a state known as the mixed state and often represented as {| ⟩; | ⟩}).
Quantum entanglement: Superposition also applies to the quantum state of two separate systems (such as two photons and two electrons). These two systems are said to be entangled if their state cannot be described separately, e.g.: which represents a superposition in which the two systems are in both state A and state B. In contrast, a separable state can always be described as the product of the independent states of systems 1 and 2, e.g.:

B: Multimode emission and mixed state in heralded single-photon sources
In SPDC and SFWM, the strong correlations between signal and idler photons determined by momentum and energy conservation can lead to multimode emission. For example, we can consider the frequency correlations between signal and idler photons, where the state can be expressed as: where frequency pairs ωn and ω-n sum up to the pump frequency for SPDC (or twice the frequency pump for SFWM): . For simplicity, we consider the case of discrete frequencies, as would be the case for SPDC or SFWM in a cavity; however, similar results would also be obtained when taking into account the continuous character of the frequency distribution. Measuring the heralding photon (say the idler) without resolving its frequency projects the signal photon into a mixed state | ⟩ = {| 1 ⟩; | 2 ⟩; | 3 ⟩; … }. A possible solution is to filter only a single frequency mode so that the state is, e.g., | ⟩ / = | 2 ⟩ | −2 ⟩ . In this case, measuring the heralding idler photon will project the single photon into the pure state | 2 ⟩ .

C: Relation between non-classical correlations and entanglement
Entanglement and non-classical correlations, which are commonly interchanged, are in fact quite different. While the presence of non-classical correlations is not enough to demonstrate entanglement, the reverse is trueentanglement guarantees the presence of non-classical correlations. To better clarify the difference, we consider a practical example such as the so-called "twin beams". In this case, as the name suggests, two beams are said to be twins if they display exactly the same intensity at the single-photon level. For example, a laser field impinging on a lossless perfectly balanced 50/50 beam splitter will not generate twin beams; indeed, the intensity at the output ports of the beam splitter is the same only on average. The two beams will exhibit non-correlated intensity fluctuations determined by the quantum nature of light (shot-noise). The amplitude of these fluctuations scales as the inverse square root of the average intensity. On the other hand, in both the SPDC and SFWM processes, in the ideal scenario, the signal and idler beams generated will exhibit the exact same photon statistics. This is intrinsic to the generation process, as one signal photon can be generated if and only if an idler photon is also generated.
Indeed, one means for proving the entanglement of a bipartite system is based on the Peres-Horodecki criterion 1,2 . It defines a necessary condition for separabilityits violation is a sufficient (but not necessary) condition for entanglement 3 . However, the violation of this criterion is a sufficient and necessary condition only for 2×2 and 2×3 systems (bipartite two-mode and bipartite tri-mode systems, respectively). We note that the Peres-Horodecki criterion is more sensitive than Bell's inequalities in the sense that there exist states that are entangled according to the Peres-Horodecki criterion that do not violate any of Bell's inequalities 3,4 .